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A299409 Solution (e(n)) of the system of 5 complementary equations in Comments. 6
10, 26, 45, 62, 78, 94, 114, 130, 146, 162, 180, 198, 214, 230, 248, 266, 282, 298, 317, 334, 350, 366, 386, 402, 418, 434, 451, 470, 486, 502, 520, 538, 554, 570, 589, 606, 622, 638, 658, 674, 690, 706, 725, 742, 758, 774, 792, 810, 826, 842, 861, 878, 894 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:

a(n) = least new;

b(n) = least new;

c(n) = least new;

d(n) = least new;

e(n) = a(n) + b(n) + c(n) + d(n);

where "least new k" means the least positive integer not yet placed.

***

Conjecture: for all n >= 0,

0 <= 17n - 11 - 4 a(n) <= 4

0 <= 17n - 7 - 4 b(n) <= 4

0 <= 17n - 3 - 4 c(n) <= 3

0 <= 17n + 1 - 4 d(n) <= 3

0 <= 17n - 5 - e(n) <= 3

***

The sequences a,b,c,d,e partition the positive integers.  The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

EXAMPLE

n:   0  1   2    3   4   5   6   7   8   9

a:   1  5   9   14  18  22  27  31  35  39

b:   2  6   11  15  19  23  28  32  36  40

c:   3  7   12  16  20  24  29  33  37  41

d:   4  8   13  17  21  25  30  34  38  42

e:  10  26  45  62  78  94 114 130 146 162

MATHEMATICA

(* Program 1: sequences a, b, c, d, e generated from the complementary equations *)

z = 200;

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,

Last[a] + Last[b] + Last[c] + Last[d]];

Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],

   AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],

   AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],

   AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],

   AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];

Take[a, 100]  (* A299405 *)

Take[b, 100]  (* A299637 *)

Take[c, 100]  (* A299638 *)

Take[d, 100]  (* A299641 *)

Take[e, 100]  (* A299409 *)

(* Program 2: sequence e generated by iterating a morphism *)

morph = Nest[Flatten[# /. Thread[{0, 1, 2, 3} -> {{2, 3, 3, 1}, {2, 3, 2, 1}, {2, 3, 1, 1}, {2, 3, 0, 1}}]] &, {0}, 9];

A299409 = Accumulate[Prepend[Drop[Flatten[morph /. Thread[{0, 1, 2, 3} -> {{1, 1, 2, 4}, {1, 1, 3, 3}, {1, 1, 4, 2}, {1, 1, 5, 1}}]], 1] + 15, 10]];

Take[A299409, 100]  (* Peter J. C. Moses, May 04 2018 *)

CROSSREFS

Cf. A036554, A299634, A299405, A299637,  A299638, A299641.

Sequence in context: A223451 A229309 A044452 * A198017 A137351 A134406

Adjacent sequences:  A299406 A299407 A299408 * A299410 A299411 A299412

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 22 2018

STATUS

approved

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Last modified August 4 16:34 EDT 2020. Contains 336202 sequences. (Running on oeis4.)